Homology classes of real algebraic sets
type: Item
instance of: scholarly article
title: Homology classes of real algebraic sets
zbMATH Open document ID: 1153.14035
author: Wojciech Kucharz
publication date: 2008-07-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10341
review text: The program of comparison between algebraic and topological categories to which the present paper contributes has its origin in \textit{J. Nash}'s celebrated 1952 paper on real algebraic manifolds [Ann. Math. (2) 56, 405--421 (1952; Zbl 0048.38501)]. It obtained a strong impetus by \textit{A. Tognoli}'s 1973 paper [Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 27, 67--185 (1973; Zbl 0263.57011)] where it is proved that every compact \(C^\infty\) submanifold of an \(\mathbb{R}^m\) is diffeomorphic to a nonsingular algebraic subset of an \(\mathbb R^p;\) that is, every such manifold has a so called algebraic model. For this see also chapter 14 of the book [BCR] by [\textit{J. Bochnak, M. Coste, M.-F. Roy}, Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36. (Berlin): Springer. (1998; Zbl 0912.14023)]. In the following we closely follow the author's introduction making use of [BCR]. Given an affine real algebraic variety \(X\) in the sense of [BCR, p. 63], denote by \(H_p(X,\mathbb{Z}/2)\) its \(p\)-th homology group with coefficients in \(\mathbb{Z}/2,\) by \(H_p^{\text{alg}}(X,\mathbb{Z}/2)\) the subgroup presented by the \(p\)-dimensional subvarieties of \(X,\) and let \(H^q_{\text{alg}}(X,\mathbb{Z}/2)\) be its inverse image under Poincaré Duality. So \(p+q=n.\) The groups \(H^q_{\text{alg}}(-,\mathbb{Z}/2)\) of algebraic cohomology classes are central in real algebraic geometry and satisfy the expected functorial property that given a regular map \(f:X\rightarrow Y\) between compact nonsingular real algebraic varieties then the induced homomorphism \(f^*: H^q(Y,\mathbb Z/2)\rightarrow H^q(X,\mathbb Z/2)\) maps the subgroup \(H^q_{\text{alg}}(Y,\mathbb{Z}/2)\) into \(H^q_{\text{alg}}(X,\mathbb{Z}/2).\) The direct sum \(H^*_{\text{alg}}(X,\mathbb{Z}/2)= \bigoplus_{q\geq 0} H^q_{\text{alg}}(X,\mathbb{Z}/2)\) is a subring of the cohomology ring sum \(H^*(X,\mathbb{Z}/2)\) and the Stiefel-Whitney class \(w_q(X)\)is in \(H^q_{\text{alg}…
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14F25
zbMATH DE Number: 5298328
zbMATH Keywords: algebraic models
zbMATH Keywords: algebraic cycles
zbMATH Keywords: regular maps
zbMATH Keywords: vector bundles
zbMATH Keywords: cohomology
MaRDI profile type: Publication
OpenAlex ID: W2473967276
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DOI: 10.5802/AIF.2376
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5298328